Properties

Label 1-73-73.7-r1-0-0
Degree $1$
Conductor $73$
Sign $0.997 - 0.0679i$
Analytic cond. $7.84493$
Root an. cond. $7.84493$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 − 0.866i)2-s i·3-s + (−0.5 + 0.866i)4-s + (−0.965 + 0.258i)5-s + (−0.866 + 0.5i)6-s + (0.707 + 0.707i)7-s + 8-s − 9-s + (0.707 + 0.707i)10-s + (0.258 + 0.965i)11-s + (0.866 + 0.5i)12-s + (0.965 + 0.258i)13-s + (0.258 − 0.965i)14-s + (0.258 + 0.965i)15-s + (−0.5 − 0.866i)16-s + (−0.707 − 0.707i)17-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)2-s i·3-s + (−0.5 + 0.866i)4-s + (−0.965 + 0.258i)5-s + (−0.866 + 0.5i)6-s + (0.707 + 0.707i)7-s + 8-s − 9-s + (0.707 + 0.707i)10-s + (0.258 + 0.965i)11-s + (0.866 + 0.5i)12-s + (0.965 + 0.258i)13-s + (0.258 − 0.965i)14-s + (0.258 + 0.965i)15-s + (−0.5 − 0.866i)16-s + (−0.707 − 0.707i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 73 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.997 - 0.0679i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 73 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.997 - 0.0679i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(73\)
Sign: $0.997 - 0.0679i$
Analytic conductor: \(7.84493\)
Root analytic conductor: \(7.84493\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{73} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 73,\ (1:\ ),\ 0.997 - 0.0679i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8843917006 + 0.03009699073i\)
\(L(\frac12)\) \(\approx\) \(0.8843917006 + 0.03009699073i\)
\(L(1)\) \(\approx\) \(0.6957503700 - 0.2381840963i\)
\(L(1)\) \(\approx\) \(0.6957503700 - 0.2381840963i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad73 \( 1 \)
good2 \( 1 + (-0.5 - 0.866i)T \)
3 \( 1 - iT \)
5 \( 1 + (-0.965 + 0.258i)T \)
7 \( 1 + (0.707 + 0.707i)T \)
11 \( 1 + (0.258 + 0.965i)T \)
13 \( 1 + (0.965 + 0.258i)T \)
17 \( 1 + (-0.707 - 0.707i)T \)
19 \( 1 + (-0.866 + 0.5i)T \)
23 \( 1 + (0.866 + 0.5i)T \)
29 \( 1 + (0.965 + 0.258i)T \)
31 \( 1 + (0.965 + 0.258i)T \)
37 \( 1 + (-0.5 + 0.866i)T \)
41 \( 1 + (0.5 - 0.866i)T \)
43 \( 1 + (-0.707 + 0.707i)T \)
47 \( 1 + (0.258 + 0.965i)T \)
53 \( 1 + (-0.258 + 0.965i)T \)
59 \( 1 + (-0.258 + 0.965i)T \)
61 \( 1 + (-0.866 - 0.5i)T \)
67 \( 1 + (0.866 - 0.5i)T \)
71 \( 1 + (0.5 + 0.866i)T \)
79 \( 1 + (-0.866 + 0.5i)T \)
83 \( 1 + (0.707 + 0.707i)T \)
89 \( 1 + (-0.5 - 0.866i)T \)
97 \( 1 - iT \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−31.69256685154389389704665396587, −30.45086287591734763343125481109, −28.53330579912489197729985180981, −27.723095698533816039422593532912, −26.87898724508447856475291774059, −26.30351027462310641068686978963, −24.76723060774371092484892252544, −23.63526178284193893116958135842, −22.92387040097028510610109892253, −21.36346168773775840331000183471, −20.09922080517630425268707084244, −19.15401365465555377406971388219, −17.499957601557947946238997678748, −16.61087366230569736864191821218, −15.651559744988601481299643354515, −14.79080875291029559033658906626, −13.52825616159151943483909076755, −11.2067008429608068841280148387, −10.594039239569802132934473387615, −8.71776901707641266631811455317, −8.255376883203085097743006008186, −6.46768784286789951794947056426, −4.844087069114438208931145620135, −3.86667442188692715377466458267, −0.59809977602160066443202796601, 1.37685065250904311620182589381, 2.791340741001799144218778662024, 4.54107521198267993873202523577, 6.81458518908115740360569877054, 8.05009998183072880654266061038, 8.90918425141482516257286505424, 10.93220068511939668271787605968, 11.78689414606800258768561079091, 12.58535625172870812269525368204, 14.02985826729847243314629617747, 15.50512990550333813240021057124, 17.26383914819916339926534793700, 18.24041891849468578695667472558, 18.98657216808285461861727288379, 19.99544044427905338541366130760, 21.06229215375930774392829492923, 22.65347881210654549470603934666, 23.40391443690179833487235313247, 24.89169222559200574116662392622, 25.8755285633908682541706699581, 27.31292648540897826794280111593, 28.065165089388778103094854650195, 29.09413846394984232305146953640, 30.41947710244288138803444804817, 30.91479526243548049257036202883

Graph of the $Z$-function along the critical line